The old Southern Region were past masters at this, but much seems to have fallen away, for a range of reasons.
Principal is modern generation rolling stock doesn't like joining and dividing. One would think it would be in the original design spec, and indeed seems possible on equivalent modern stock overseas, but not here. I think the nadir was the Class 458, which needed (needs?) half an hour to reboot everything. Old generation stock could divide and the first half be away in one minute, with the second section following onto a different route just a further minute later.
It seems the need for a separate staff of shunters on site to handle the attaching has been removed as a cost saving.
On time reporting is impacted by difficulties where one of the sections is delayed. Easier to run separate trains. If the connection is missed, too bad. Missed connections are not reported to the DfT, or to Roger Ford for his Golden Whistles.
Southern still seem to do it, though less so than in around 2013.
Also SWR still do it at Waterloo (in the manner described above for Epsom/Caterham/Tattenham services), or certainly did in the 2004-19 period - and at Bournemouth (front 5 to Weymouth).
As you say more time is needed nowadays for a split-join process. In the 80s typically a train would arrive, the front portion would be away in 2 min, and the rear portion in 4 (I never saw any examples of 3 mins as presumably there is a minimum 2-minute headway between departures from the same platform). One example was the Basingstoke/Alton dividers. In some hours they would arrive Woking xx11/41, Basingstoke portion out xx13/xx43 and Alton portion out xx15/xx45. This was only possible, however, in hours when there was no xx14/xx44 towards Portsmouth, which would have to be slotted in between the xx13 and xx15 (easy to see why, with a conflict, that would not work too well). Otherwise the Alton left at xx17/xx47.
Now it seems to be front portion away in 4 mins and rear portion in 7, so significantly less efficient and losing more time.
With this lower efficiency you can see why it's not as attractive as it once was.
Re the separate staff of shunters, how did they manage when they did splitting and joining at remote termini such as East Grinstead and Tattenham Corner? At such places it was common practice to leave one unit there during the off-peak period and pick it up again in time for the evening peak.
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